Bergman metric explained

In differential geometry, the Bergman metric is a Hermitian metric that can be defined on certain types of complex manifold. It is so called because it is derived from the Bergman kernel, both of which are named after Stefan Bergman.

Definition

Let

G\subset{C

}^n be a domain and let

K(z,w)

be the Bergman kernelon G. We define a Hermitian metric on the tangent bundle

T_z{C

}^n by

g_ij(z) :=

	\partial²
	\partialz_i\partial\bar{z

_j}
logK(z,z),

for

z\inG

. Then the length of a tangent vector

\xi\inT_z{C

}^n isgiven by

\left\vert\xi\right\vert_B,z

	n
:=\sqrt{\sum
	i,j=1

g_ij(z)\xi_i\bar{\xi}_j}.

This metric is called the Bergman metric on G.

\gamma\colon[0,1]\to{C

}^n isthen computed as

\ell(\gamma)

	1
= \int
	0

\left\vert

	\partial\gamma
	\partialt

(t)\right\vert_B,\gamma(t)dt.

The distance

d_G(p,q)

of two points

p,q\inG

is then defined as

d_G(p,q):=
inf\{\ell(\gamma)\midallpiecewiseC^{1curves\gammasuchthat\gamma(0)=pand\gamma(1)=q}\}.

The distance d_G is called the Bergman distance.

The Bergman metric is in fact a positive definite matrix at each point if G is a bounded domain. More importantly, the distance d_G is invariant underbiholomorphic mappings of G to another domain

. That is if fis a biholomorphism of G and

, then

d_G(p,q)=d_G'(f(p),f(q))

References

Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.